Quant Boosters  Rajesh Balasubramanian  Set 1

Total number of subsets =
$2^{10}$
For choosing any subset, each element can either be part of the subset or not part of the subset. So, for each element, there are two options. So, with 10 elements in the set, we can create$2^{10}$ subsets. We should bear in mind that this$2^{10}$ includes the 2 improper subsets as well. The whole set P and the null set are included in this$2^{10}$ .
Subsets whose product is not a multiple of 3 = Subsets of the set {1, 2, 4, 5, 7, 8, 10} =$2^{7}$ . This includes the empty subset also. So, the correct answer should be$2^{7}$ –1

Q11) A, B, C, D, E are doctors, P, Q, R, S, are engineers. In how many ways can we select a committee of 5 that has more engineers than doctors?

Two scenarios are possible.
3 engineers and 2 doctors: 4C3 × 5C2 = 4 × 10 = 40
4 engineers and 1 doctor : 4C4 × 5C1 = 1 × 5 = 5
Total number of possibilities = 40 + 5 = 45

Q12) From a card pack of 52, in how many ways can we pick a sequence of 4 cards such that they are in order and from different suits? Consider Ace to be the card following King in each suit. So, Ace can be taken to precede ‘2’ and succeed ‘King’. So, JQKA would be a sequence, so would be A234. However, QKA2 is not a sequence.

4 cards in order can be A234, 2345, ….JQKA. 11 different possibilities
For a given set of four cards, say 2345, they can be from 4 different suits in 4! ways.
So, total number of possibilities = 11 × 4! = 264.

Q13) In how many ways can letters of the word ATTITUDE be rearranged such that no two T’s are adjacent to each other?

ATTITUDE has 8 letters, of which 3 are T’s
Now, Let us place the letters that are not Ts on a straight line. We have AIUDE. These can be arranged in 5! Ways. Now let us create slots between these letters to place the Ts in. In order to ensure that no two T’s are adjacent to each other, let us create exactly one slot between any two letters.
A __ I __ U __ D __ E.
Additionally, let us add one slot at the beginning and end as well as the T’s can go there also
A __ I __ U __ D __ E
Now, out of these 6 slots, some 3 can be T. That can be selected in 6C3 ways.
So, total number of words = 5! × 6C3 = 2400

Q14) In how many rearrangements of the word SLEEPLESS will no two S’s appear together?

Let us aggregate the other letters LEEPLE and arrange these.
These can be arranged in 6!/2!3! ways. Now, let us create slots in between and before/after these letters, where S can potentially appear.
__ L __ E __ E __ P __ L __ E__.
Out of these 7 slots, 3 should be taken up by S’s. This can be done in 7C3 ways.
So, total number of rearrangements = 6!/2!3! × 7C3 = 60 × 35 = 2100

Q15) How many numbers of up to 5 digits can be created using the digits 1, 2 and 3 that are multiples of 12?

For a number to be a multiple of 12, it has to be a multiple of 3 and of 4. So, the last two digits have to be a multiple of 4 and the sum of digits should be a multiple of 3.
We need to break this down by number of digits.
2–digit number: Only possibility 12.
3–digit numbers: These can end in 12 or 32. If it ends in 12, the sum of these two digits is 3, the only value the first digit can take is 3. Similarly if it ends in 32, the only value the first digit can take is 1. So, two 3–digit numbers are possible 312 and 132.
4–digit numbers: Again, these can in 12 or 32.
Scenario I: Ending in 12. Sum of these two digits = 3. First two digits can be 12 or 21 or 33.
Scenario II: Ending in 32. Sum of these two digits = 5. First two digits can be 22, 13 or 31.
So, possible 4–digit numbers are 1212, 2112, 3312, 2232, 1332, 3132.
5–digit numbers: Again, these can end in 12 or 32.
Scenario I: Ending in 12. Sum of these two digits = 3. First three digits can be 111, 222, 333, 123 (6 rearrangements 132, 213, 231, 312, 321) – 9 possibilities
Scenario II: Ending in 32. Sum of these two digits = 5. First three digits can add up to 4 or 7.
First 3–digits adding up to 4: 112, 121, 211. 3 possibilities
First 3–digits adding up to 7: 223, 232, 322, 133, 313, 331. 6 possibilities
9 possibilities overall
Total number of 5–digit numbers = 9 + 9 = 18
Overall, total number of possibilities = 1 + 2 + 6 + 18 = 27 numbers.

Q16) How many numbers of up to 5 digits can be created using the digits 1, 2, 3, 5 each at least once such that they are multiples of 15?

For a number to be a multiple of 15, it has to be a multiple of 3 and of 5. So, the last digit has to be 5 and the sum of digits should be a multiple of 3.
We can have either 4–digit or 5–digit numbers. If we have a 4–digit number, sum of the digits will be 1 + 2 + 3 + 5 = 11. No 4–digit number formed with digits 1, 2, 3, 5 exactly once can be a multiple of 3. So, there is no possible 4–digit number.
Now, in any 5 digit number, we will have 1, 2, 3, 5 once and one of these 4 digits repeating once. 1 + 2 + 3 + 5 = 11. So, the digit that repeats in order for the number to be a multiple of 3 has to be 1. In this instance, sum of the digits will be 12 and this is the only possibility.
So, any 5–digit number has to have the digits 1, 1, 2, 3, 5. For the number to be a multiple of 5, it has to end in 5.
So, number should be of the form __ __ __ __ 5, with the first 4 slots taken up by 1, 1, 2, 3.
These can be rearranged in 4!/2! = 12 ways.
There are 12 possibilities overall.

Q7) How many 4–digit numbers with distinct digits exist product of whose digits is a non–zero multiple of 9?

For the product to be zero, one of the digits has to be zero. So, if the product is non–zero, no digit can be zero.
For the product to be a multiple of 9, one of the digits can be 9, or we could have 3 and 6 as two digits of the number.
Scenario I: One of the digits being equal to 9: 9 can be either in the 1000’s place, or 100’s place, or 10’s place or units place. Now, with 9 in the 1000’s place, we will have 8 × 7 × 6 numbers totally.
So, overall there will be 8 × 7 × 6 × 4 numbers possible. 56 × 24 = 1344 numbers
Scenario II: Two of the digits being equal to 3 and 6: Now, in this list we should exclude all numbers that have a 9 as we would have already accounted for these. So, we need to count all possibilities where two digits being 3 and 6, other two selected from 1, 2, 4, 5, 7, 8. No of ways of selecting 2 digits out of 6 = 6C2.
Number of ways of rearranging = 4!.
So, number of numbers = 6C2 × 4! = 15 × 24 = 360
Total number of possible numbers = 1344 + 360 = 1704.

Q18) From 4 doctors, 3 engineers and 5 scientists, in how many ways can we create a committee of 6 to 8 people that has more scientists than engineers, more engineers than doctors, and at least one doctor?

Three scenarios are possible – the committee can have 6, 7 or 8 people
Scenario I: 6–member committee. 3 scientists,
2 engineers and 1 doctor. Number of ways =
5C3 × 3C2 × 4C1 = 10 × 3 × 4 = 120
Scenario II: 7–member committee.
4 scientists, 2 engineers and 1 doctor: Number of ways = 5C4 × 3C2 × 4C1 = 5 × 3 × 4 = 60
Scenario III: 8–member committee.
5 scientists, 2 engineers and 1 doctor: Number of ways = 5C5 × 3C2 × 4C1 = 1 × 3 × 4 = 12
4 scientists, 3 engineers and 1 doctor: Number of ways = 5C4 × 3C3 × 4C1 = 5 × 1 × 4 = 20 ways
Total number of possibilities = 120 + 60 + 12 + 20 = 212.

Q19) A flag is formed with 4 vertical bands. If we can choose from colors blue, green, and yellow and no two adjacent bands should have the same color, how many different flags can be created?

Scenario I: 2 colours chosen: Number of ways of selecting 2 colours = 3C2 . For the two colours that have been selected, say A and B, there are two arrangements possible – ABAB or BABA. Total number of flags with 2 colors = 3C2 × 2 = 3 × 2 = 6
Scenario II: All 3 colours being chosen. Any one colour will have to be repeated. Selecting the one colour that repeats can be done in 3C1 ways. Post this, we have AABC. This can be rearranged in 4!/2! ways. Of these there will be 3! ways when the two A’s appear together. So, the number of ways where 2 A’s do not appear together will be 12 – 6 = 6. These are ABAC, ABCA, ACAB, ACBA, BACA, CABA. Total number of possible flags where three colours are chosen = 3C1 × 6 = 18 ways
So, there are 6 + 18 = 24 different flags possible totally.
This can also be done with another approach.
Let the four bands be called ABCD.
A can be selected in 3 ways. It could be blue, green or yellow.
B can be any of the three colours except A. So, there are 2 possible options for B.
C can be any of the three colours except B. So, there are 2 possible options for C.
D can be any of the three colours except C. So, there are 2 possible options for D.
Total number of options = 3 × 2 × 2 × 2 = 24 ways

Q20) An equation ax^2 + 8x + c = 0 has two distinct real roots. If a, c are positive integers less than 11, how many values can (a, c) take?